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why does index futures swing (in absolute) more than index, when index futures price is lower than index (Backwardation)?

Say, SET50 Index(Thailand) is at 950, SET50 active Futures will be at around 945 (1 month to expiration). If SET50 moves 10 points, SET50 active Futures will, on average, move more than 10 points.(by comparing (Daily and intra-day standard deviation of $\Delta S$ and $\Delta F$)

  1. What is an explanation?

Given that $F=Se^{(r-q)(T-t)}$ and $\Delta _F=e^{(r-q)(T-t)}$ and minimum variance hedge ratio($h^*$) = $\rho \frac {\Delta S}{\Delta F}$,

  1. Is there any relationship between $\Delta _F$ and $h^*$?

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If I want to hedge SET50 with SET50 Futures (that swing more than index) and minimise basis risk, what Delta should I use if

  1. I want to hedge until futures expiration

  2. I want to close out hedge position before expiration (say, 5 days) and futures prices are mostly traded fair

  3. I want to close out hedge position before expiration (say, 5 days) and futures prices are always traded cheap but expected to be converged to fair at 5 day before expiration

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The empirical relationship between the futures price $F$ and the spot price $S$ is

$$ F = S e^{b\tau} $$

where $\tau$ is the time to expiry, and $b$ is the empirical basis, i.e. the number that makes the equation hold, given by

$$ b = \frac{1}{\tau}\log(F/S) $$

It can be compared to the theoretical basis,

$$ b_{\rm theor} = r - q $$

where $r$ is the funding rate and $q$ is the expected dividend rate to expiry, but in general you will have $b\neq b_{\rm theor}$ (due to transaction costs, regulation, taxes and other limits to arbitrage).

The percentage futures return can be expressed as

$$ \frac{\delta F}{F} \approx \frac{\delta S}{S} + \tau \cdot \delta b + b \cdot \delta\tau $$

or, with $r_f=\delta F/F$ and $r_s = \delta S/S$,

$$ r_f\approx r_s+ \tau \cdot \delta b + b \cdot \delta\tau $$

The futures delta is, as you said,

$$ \Delta_F = e^{b\tau} $$

which is, in general, less than one if the futures are in backwardation (i.e. $b < 0$) which is true about 75% of the time for SET50 futures. However, as you correctly point out, when the spot moves by 10 points, the futures tend to move by more than 10 points, not less (a rough calculation suggests that the futures move by around 10.5 points for every 10 point move in the underlier).

If you want to hedge the futures by holding an offsetting amount of the spot, one option is to hold $\Delta_F$ of the spot. Alternatively you can take into account the comovement of the futures and the spot, and compute the hedge ratio $\beta$ to minimize the square of $r_f - \beta \cdot r_s$ (note that here we are talking in percentage terms, rather than absolute terms - to convert the hedge ratio back to absolute terms, you should multiply by $\Delta_F$, i.e. you would hold $\beta\cdot\Delta_F$ of the spot for each unit of the futures).

$$ \begin{align} \beta & = \frac{{\rm Cov}(r_f,r_s)}{{\rm Var}(r_s)} \\ & \approx \frac{{\rm Var}(r_s) + \tau \cdot {\rm Cov}(r_s, \delta b)}{{\rm Var}(r_s)} \\ & = 1 + \tau \frac{{\rm Cov}(r_s, \delta b)}{{\rm Var}(r_s)} \\ & = 1 + \tau \cdot \rho_{b,s} \frac{\sigma_b}{\sigma_s} \end{align} $$

where $\rho_{b,s}$ is the correlation between changes in the empirical basis and changes in the spot, and $\sigma_b$, $\sigma_s$ are the volatilities of the basis and the spot.

This tells us that the hedge ratio will be greater than one if changes in the spot are positively correlated with changes in the empirical basis, and less than one if changes in the spot are negatively correlated with changes in the empirical basis.

I measure a daily correlation of around 0.05 between changes in the spot and changes in the basis, indicating that you should over-hedge the futures with the spot, as opposed to a hedge ratio of 1 that you would use if you didn't take spot/basis correlation into account.

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  • $\begingroup$ Thank for your detailed answer! Shouldn't I need to minimize the square of $\delta f −β⋅delta_s$ instead? I mean I want to reduce daily P&L change of combined hedged position(Futures and Index) $\endgroup$ – Jack JackGuRae Jan 18 '17 at 11:04
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    $\begingroup$ You can do that instead. I find it more intuitive to minimize the difference in returns. You will just end up with an extra factor of $\Delta_F$ in your hedge ratio, as I alluded to in the paragraph where I talked about the difference between absolute and percentage returns). $\endgroup$ – Chris Taylor Jan 18 '17 at 11:07
  • $\begingroup$ So $\Delta_F =e^{b\tau}$ doesn't work well if I want to minimize daily P&L change of combined hedged position? $\endgroup$ – Jack JackGuRae Jan 18 '17 at 11:16
  • $\begingroup$ It gives an optimal hedge if the changes in the empirical basis are uncorrelated with the changes in the spot index. If there is a correlation, the hedge is not optimal. In practice there is a small positive correlation, implying that you would want to over-hedge (i.e. hold a bit more than $\Delta_F$ of the spot for each unit of futures). $\endgroup$ – Chris Taylor Jan 18 '17 at 12:27

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